Here we consider
geometry of the Universe on large scales,
and leave the question of topology for a future investigation.
Namely, to each geometry one can
associate different topologies. For example,
if the Universe's spatial section are flat, then its geometry
is \({R}^3\) , which can be considered as the covering
space of various topologies, the simplest ones being
toroidal (\({T}^3\) ), cuboidal
(\({T}^2\times {R}\) ) and slab
(\({T}\times {R}^2\simeq S^1\times {R}^2\) ).
There is a large body of
work devoted to investigating topology
of the Universe, for reviews of existing attempts
see Refs. [1]}, [2]}, [3]}, [4]}, [5]}.
Early attempts include the works of Starobinsky [6]}
and de Oliveira-Costa and Smoot [7]},
where the authors use the COBE data to
investigate whether the Universe has toroidal topology,
and find no evidence for it. Refs. [8]}, [9]} look for circles in the CMB sky that would be an
important sign of topology, but find no compelling evidence
in favor of such circles.
